Let theta be a (finite) real number, and show that if (a_n) is a sequence such that the limit (as n goes to infinity) of sup a_n is theta, and if (a_n) has a subsequence that converges to a limit L, then L is less than or equal to theta.
Please see the attached.
This question concerns the limits that subsequences of a given infinite sequence of real numbers could converge to, given the condition on the limit superior (lim sup) of the entire sequence. A complete, detailed proof of the stated result, including the definition of a key term used in the question and the solution, is provided. The solution is presented in a .pdf file that contains nearly 200 words.